Question

If $A$ and $B$ are non-singular square matrices of even order such that $A^{T}=A,B^{T}=-B$ and $AB+BA=O$
(where $O$ is a null matrix ), then

• A. $\left(A B\right)$ is skew symmetric matrix
• B. $A^{- 1}$ and $B^{- 1}$ commute each other
• C. $\left(A^{- 1} B\right)$ is symmetric matrix
• D. $B$ and $A^{- 1}$ commute each other

Answer 1

Answer: (C)

$AB=-BA\Rightarrow A^{- 1}AB=-A^{- 1}BA$
$\Rightarrow B=-A^{- 1}BA\Rightarrow BA^{- 1}=-A^{- 1}B$
$\left( A^{- 1} B\right)^{T}=B^{\top}\left(A^{\top}\right)^{- 1}=-BA^{- 1}=A^{- 1}B$